1. Computational Methods for Kinetic Processes in Plasma Physics
Ken Nishikawa
National Space Science & Technology Center/UAH
Basics of PIC Simulation methods
* Collisionless plasmas
* Finite-size particles
* Electrostatic codes
* Charge assignment and force interpolation (already in 3-D system)
* Filtering action of shape function
* Summary
September 1, 2011

2. Context Collisionless plasmas Finite-size particles
Electrostatic codes
Charge assignment and force interpolation
(already in 3-D system)
Filtering action of shape function
Summary

4. Collisionless plasmas
Number of particles in Debye cubes

5. Collisionless plasma can be described by Vlasov-Maxwell equations
with distribution function f(x, v, t) (6 dimensions):
Particle method
Direct calculation of this set of equations – 6D
Improvements have been made, but difficult
to calculate using this method

6. Coulomb force and force law
F=Q/r2 (3D) F
F=Q/r (2D)
Coulomb
collision
is reduced
using
finite-size
particles
a: radius of particle
(collective effects)
(Dawson 1983)

7. Particle mover accuracy: Simple harmonic motion test

8. Two major methods of calculating current
Spectral method (UPIC code) (note by Decyk)
We will review this method in details later after we do handout
exercises
2. Charge-conserving current deposit (Villasenor & Buneman 1992)
We will review this method with Umeda’s method later

9. Electrostatic codes Four major criteria to choose an Algorism
Time scales of the system >> light crossing time, static magnetic field
Four major criteria to choose an Algorism
for integration of equation of motion
Convergence: the numerical solution converges
to the exact solution of the differential equation
in the limit of Δt and Δx tend to zero
Accuracy: the truncation error associated with
approximating derivatives with differences
Stability: depends on how total errors (including
truncation error and round-off errors) grows in time
Efficiency: the code needs to be efficient to handle
large number of particles
Need asses two physical quantities to know how well the codes work
Dissipation: The truncation error associated with approximating derivatives with differences causes the dissipation of some physical quantities
Conservation: The truncation error also causes the deviation of the conservation law

10. Integration of equations of motion
The simple second order leapfrog achieves the best balanced between accuracy
stability, and efficiency

11. Charge assignment and force evaluation by cloud-in-cell in 1D
As assigned in 3D system the same interpolation scheme is used in 1D

12. Density assignment in 3D system (2D)
c for electrons
do 3 n0=1,lecs
i=x(n0)
dx=x(n0)-i
cx=1.-dx
j=y(n0)
dy=y(n0)-j
cy=1.-dy
k=z(n0)
dz=z(n0)-k
cz=1.-dz
C Smoothing with the (.25,.5,.25) profile in each dimension:
sl=.5
do 121 l=-1,1
sl=.75-sl
sr=.5
do 121 m=-1,1
sr=.75-sr
sn=.5
do 121 n=-1,1
sn=.75-sn
s=sl*sr*sn
rhe(i+l ,j+m ,k+n )=rhe(i+l ,j+m ,k+n ) s*cx*cy*cz
rhe(i+l+1,j+m ,k+n )=rhe(i+l+1,j+m ,k+n )+s*dx*cy*cz
rhe(i+l ,j+m+1,k+n )=rhe(i+l ,j+m+1,k+n )+s*cx*dy*cz
rhe(i+l+1,j+m+1,k+n )=rhe(i+l+1,j+m+1,k+n )+s*dx*dy*cz
(x(n0),y(n0))
l=-1, m=-1
l=0, m=-1
l=1, m=-1
l=1, m=1

13. PIC Approach to Vlasov Equation
Lorentz-Force:
Solving Maxwell equations on grid
Charge assignment
(conserving charge current)
Force Interpolation

14. Current deposit scheme (2-D)y x

15. Ampere equation
In Yee Lattice ex, ey, ez, bx, by, bz are, respectively staggered and shifted on 0.5 from (I, j, k) and located at the position
Yee lattice

16. Field update
Here

17. Electric field update

18. Particle update Newton-Lorentz equation Buneman-Boris method
Half an electric acceleration
Pure magnetic rotation
Another half electric acceleration

19. Buneman-Boris method
rotation

20. Buneman-Boris method (cont)
4 steps

21. Relativistic generalization

22. Force interpretations
“volume” weight
on (x, j, k)

23. similarly on (x, j+1, k), (x, j, k+1), (x, j+1, k+1)

24. Current deposit
Charge conservation
Villasenor and Buneman 1992

25. Schematic computational cycle
(Cai et al. 2006)

26. Time evolution of RPIC code

27. Code development Combine these components
Set initial conditions for each problem you would like to investigate
Apply MPI for speed-up
Develop graphics using NCARGraphic, AVSExpress, IDL, etc
Analyze simulation results and compare with theory and other simulation results
Prepare report